I think they avoid teaching BC breathing is because by the time you've realized that you're out of air, you switch to your BC supply, and then take a breath, you could have spent that time swimming up 20-30 feet. When you couple that with the "Boyle breath" you can steal as you're ascending, it is likely counterproductive to your survival to breathe your BC air. You've got MAYBE one breath out of that, and it's of dubious quality. I also tend to dive with my BC deflated, so there may not even be air in there in the first place, and now you've lost precious seconds. Moreover, you're pulling an unknown amount of air out of your BC -- your ascent will be slower. What if you hit negative buoyancy because you took too much air out of your BC?
But, bad air is still air. As they say in land survival, if you're going to die of thirst, drink the dubious water. They can cure bad water; they can't cure dead. So technically, I suppose breathing from your BC would work. The problem is that if you're taking too long to fark around with your BC, you're delaying your switch to atmospheric, and that's your top priority.
They didn't even teach buddy breathing in my OW class. Our instructor said, "by the time you're doing this, you should either CESA or ditch your weights."